The existence is proved in [1]} when \(h_-\)  and \(h_+\)  have constant curvature \(K\in [-1,0)\) . The uniqueness however remains elusive. (Note that the upper bound on the curvature in Question REF  could be taken to be larger than 0, we voluntarily state the question in a relatively restricted setting.) This existence result therefore includes the case of pleated surfaces, corresponding to \(K=-1\) .
